🤖 AI Summary
Traditional algebraic rewriting is unreliable for expressions involving measurements due to domain inconsistencies arising from repeated observations and division operations. This work proposes a unified semantic framework that simultaneously tracks both the provenance and definedness of expressions, enabling sound one-way rewriting and interchangeability judgments. By introducing label-sensitive bracketing semantics, admissible domain refinement, and a relative variant of support sets, the authors develop domain-safe rewriting rules and formally prove restoration and strictness theorems. All results are fully formalized in Lean 4 without any use of `sorry`, revealing fundamental limitations: simplifications are generally irreversible, equivalence over a common domain is insufficient, and label erasure inherently causes information loss.
📝 Abstract
Ordinary algebraic equality is not a sound rewrite principle for measurement-bearing expressions. Reuse of the same observation matters, and division can make algebraically equal forms differ on where they are defined. We present a unified semantics that tracks both provenance and definedness. Token-sensitive enclosure semantics yields judgments for one-way rewriting and interchangeability. An admissible-domain refinement yields a domain-safe rewrite judgment, and support-relative variants connect local and global admissibility. Reduction theorems recover the enclosure-based theory on universally admissible supports. Recovery theorems internalize cancellation, background subtraction, and positive-interval self-division. Strictness theorems show that reachable singularities make simplification one-way and make common-domain equality too weak for licensed replacement. An insufficiency theorem shows that erasing token identity collapses distinctions that definedness alone cannot recover. All definitions and theorems are formalized in sorry-free Lean 4.